Theorem konigsbergiedgw 27108

Description: The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

Hypotheses

Ref	Expression
konigsberg.v	⊢ 𝑉 = (0...3)
konigsberg.e	⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
konigsberg.g	⊢ 𝐺 = ⟨𝑉, 𝐸⟩

Assertion

Ref	Expression
konigsbergiedgw	⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}

Distinct variable group:   𝑥,𝑉

Allowed substitution hints:   𝐸(𝑥)   𝐺(𝑥)

Proof of Theorem konigsbergiedgw

Step	Hyp	Ref	Expression
1		3nn0 11310	. . . . . . 7 ⊢ 3 ∈ ℕ0
2		0elfz 12436	. . . . . . 7 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ 0 ∈ (0...3)
4		1nn0 11308	. . . . . . 7 ⊢ 1 ∈ ℕ0
5		1le3 11244	. . . . . . 7 ⊢ 1 ≤ 3
6		elfz2nn0 12431	. . . . . . 7 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
7	4, 1, 5, 6	mpbir3an 1244	. . . . . 6 ⊢ 1 ∈ (0...3)
8		0ne1 11088	. . . . . 6 ⊢ 0 ≠ 1
9	3, 7, 8	umgrbi 25996	. . . . 5 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
10	9	a1i 11	. . . 4 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
11		2nn0 11309	. . . . . . 7 ⊢ 2 ∈ ℕ0
12		2re 11090	. . . . . . . 8 ⊢ 2 ∈ ℝ
13		3re 11094	. . . . . . . 8 ⊢ 3 ∈ ℝ
14		2lt3 11195	. . . . . . . 8 ⊢ 2 < 3
15	12, 13, 14	ltleii 10160	. . . . . . 7 ⊢ 2 ≤ 3
16		elfz2nn0 12431	. . . . . . 7 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
17	11, 1, 15, 16	mpbir3an 1244	. . . . . 6 ⊢ 2 ∈ (0...3)
18		0ne2 11239	. . . . . 6 ⊢ 0 ≠ 2
19	3, 17, 18	umgrbi 25996	. . . . 5 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
20	19	a1i 11	. . . 4 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
21		nn0fz0 12437	. . . . . . 7 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
22	1, 21	mpbi 220	. . . . . 6 ⊢ 3 ∈ (0...3)
23		3ne0 11115	. . . . . . 7 ⊢ 3 ≠ 0
24	23	necomi 2848	. . . . . 6 ⊢ 0 ≠ 3
25	3, 22, 24	umgrbi 25996	. . . . 5 ⊢ {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
26	25	a1i 11	. . . 4 ⊢ (⊤ → {0, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
27		1ne2 11240	. . . . . 6 ⊢ 1 ≠ 2
28	7, 17, 27	umgrbi 25996	. . . . 5 ⊢ {1, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
29	28	a1i 11	. . . 4 ⊢ (⊤ → {1, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
30	12, 14	ltneii 10150	. . . . . 6 ⊢ 2 ≠ 3
31	17, 22, 30	umgrbi 25996	. . . . 5 ⊢ {2, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
32	31	a1i 11	. . . 4 ⊢ (⊤ → {2, 3} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
33	10, 20, 26, 29, 29, 32, 32	s7cld 13621	. . 3 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
34	33	trud 1493	. 2 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
35		konigsberg.e	. 2 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
36		konigsberg.v	. . . . 5 ⊢ 𝑉 = (0...3)
37	36	pweqi 4162	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (0...3)
38		rabeq 3192	. . . 4 ⊢ (𝒫 𝑉 = 𝒫 (0...3) → {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2})
39	37, 38	ax-mp 5	. . 3 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
40	39	wrdeqi 13328	. 2 ⊢ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = Word {𝑥 ∈ 𝒫 (0...3) ∣ (#‘𝑥) = 2}
41	34, 35, 40	3eltr4i 2714	1 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}

Syntax hints:   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  {crab 2916  𝒫 cpw 4158  {cpr 4179  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   ≤ cle 10075  2c2 11070  3c3 11071  ℕ₀cn0 11292  ...cfz 12326  #chash 13117  Word cword 13291  ⟨“cs7 13591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595  df-s5 13596  df-s6 13597  df-s7 13598

This theorem is referenced by:  konigsbergssiedgwpr  27110  konigsbergumgr  27112